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On Klein & Milgram, concluded
Posted:
Aug 8, 2000 1:04 PM
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I have just a few more comments about the paper on long division by David
Klein and Jim Milgram.
1. cost-benefit analysis.
K & M focus on the benefits of teaching the standard long division
algorithm. I believe they exaggerate the necessity of standard long
division as the sole means to achieve several of the benefits they cite. I
also believe that many of the benefits they outline are rarely achieved in
practice. They are theoretical, not real, goods. (Interestingly, the
practical justification for teaching long division has been rendered
largely obsolete by calculators. Anyone who has a practical need to do lots
of long division problems is very unlikely to use any paper-and-pencil
algorithm. K & M had to look elsewhere for reasons to retain the algorithm.)
But even if K & M are entirely correct in their claims about the benefits
of teaching standard long division, their analysis is incomplete because
they have failed to examine the costs of teaching this algorithm. (There is
some arm waving about methods for teaching long division to children, but,
as far as I know, these ideas are speculative and have not been tested in
real classrooms. People have been trying to figure out a good way to teach
standard long division for a long time, and my guess is that if there were
one it would have been found out by now.)
It is impossible to judge whether it is worthwhile to teach the standard
long division algorithm unless the costs are examined. More generally,
other approaches to long division, with their costs and benefits, should
also be examined. And this examination should not be purely a priori. Data,
including student achievement data, should be considered.
I claim K & M seriously understimate the costs of teaching the standard
long division algorithm. I am not talking about the costs when the
algorithm is taught by an expert teacher under ideal conditions. I am
talking about the costs when ordinary teachers teach the algorithm under
typical conditions. Bringing all children to something like mastery takes
months of time over several years. The approach typically taken to the
standard long division algorithm is highly procedural and syntactic -- in
sharp contrast to the conceptual approach K & M clearly favor -- which
engenders misunderstanding of what mathematics is and consequent distaste
for the subject.
Perhaps K & M meant their piece as a correction to reformers like Leinwand
who have spent a lot of time identifying the high costs of teaching
standard long division and other paper-and-pencil algorithms. If so, I
believe they have over-corrected. A balanced assessment of costs and
benefits would have been more helpful.
2. mathematics and mathematics education.
It would be helpful if K & M could at least pretend not to think that all
math educators are idiots. It may be difficult for them to do this, but
perhaps an atmosphere of mutual respect could be established that
facilitate progress in our field.
Roughly 100 years ago, John Dewey pointed out that it's not enough to
consider the subject matter that is being taught. One must also take the
student into account. No doubt K & M have vastly greater knowledge of
mathematics than most people in mathematics education. Certainly they know
more mathematics than I do. But there are people in mathematics education
who not only know some mathematics, but have actually taught school. And
done research and read others' research. And written elementary school math
textbooks. And know the history of mathematics education. The views of such
people are not irrelevant to the questions whether and how long division
should be taught. If K & M are uninterested in the discipline of
mathematics education, I don't blame them. All I ask is that they recognize
the limits of their expertise.
3. single-issue politics.
Probably what disturbs me most is the focus on long division as the sine
qua non for a curriculum to be acceptable. Not only must paper-and-pencil
long division be taught, but one certain algorithm must be taught. Any
alternative is anathema.
I think our field is too complex to be reduced to such a litmus test.
Andy
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